I believe that anyone with a sense of smell and an interest in wine can become a wine taster, and that it takes hardly anytime at all. I am bolstered in this belief by years of listening to professionals trying to describe the same wines and doing it with completely different and even contradictory words. In practice, applying words to wine is a complete free-for-all.
No one other than you can ever know exactly how a wine strikes your senses. When it comes to wine tasting there are no absolutes. And so the opinion of the novice is every bit as valuable as that of the expert. The only difference is that the expert has been allowed to gain self-confidence, so we propound our theories rather more loudly than most newcomers.
In fact I often find that novice tasters are much better at coming up with the perfect word to describe a wine flavour than us professionals who used up our tasting vocabulary years ago. (debutants can even be better at blind tasting than professionals, partly because they have tasted fewer confusing exceptions to the rule, and also because less is expected of them).
Wine tasting is the definitive subjective sport. Once you have consciously tasted a few wines, you can build on that experience by starting to notice the common characteristics of the wine you like. Putting that together with the profiles of different grape varieties (in most cases the dominant factor in shaping how a wine tastes).